Pfaffian Phase Portrait

OBSERVATION 4 chain-order-3 towers · same chain order, very different defining integrands · slide x to see why Airy is the verification outlier
All four towers have defining integrals of the form f(x) = ∫ K(x, t) dt with the kernel K shown below. As the parameter x sweeps, watch the integrand. T_Ai is the only tower whose integrand has a polynomial-of-degree-3 phase — that's why mpmath.quadosc converges to 1e-31 on the other three but only 1e-7 on Airy. Chain order alone doesn't predict numerical-verification difficulty; phase polynomial degree does.
try x ≈ −6 to see Airy's cubic-phase oscillation explode while the other three stay calm
T_erf · chain 2 · Gaussian decay
K(x, t) = exp(−x · t²)
support: [0, x] · phase: none (pure decay) · max relerr: ~1e-31
T_Si · chain 3 · linear-phase oscillator
K(x, t) = sin(x · t) / t
support: [0, x] (finite) · phase polynomial degree: 1 (linear) · max relerr: ~1e-30
T_J · chain 3 · finite-arc oscillator
K(x, t) = cos(x · sin(t))
support: [0, π] (finite) · phase polynomial degree: 1 (linear over arc) · max relerr: ~1e-31
T_Ai · chain 3 · CUBIC-PHASE OSCILLATOR (the outlier)
K(x, t) = cos(t³/3 + x · t)
support: [0, ∞) infinite · phase polynomial degree: 3 (cubic — unique) · max relerr: 1.5e-7 via Ai″ = x·Ai ODE

The classifier: a tower's defining integral converges via standard mpmath.quadosc iff it has either finite support (T_erf, T_Si, T_J, T_K) or real-decay integrand (T_Γ). T_Ai has neither: infinite support × non-linear phase. As t → ∞, the integrand cos(t³/3 + xt) oscillates at frequency t² + x — faster than any linear-phase oscillator. mpmath.quadosc was designed assuming oscillation frequency is bounded or grows linearly; it can't adaptively shrink its step at the rate the cubic phase demands.

Stationary-phase asymptotics: Ai(x) ~ |x|−1/4 · sin((2/3)·|x|3/2 + π/4) as x → −∞. Power-decay × cubic-phase oscillator. Bessel J0 by contrast is ~ √(2/(πx)) · cos(x − π/4) — linear phase. The cubic phase is fundamentally different — it's a chain-order-3 outlier on a separate axis.

Source: airy-cubic-phase-2026-04-27 · E-201 atlas · monogate.org/superbest